# 7slides - Chapter 7: Proof Systems: Soundness and...

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Unformatted text preview: Chapter 7: Proof Systems: Soundness and Completeness Introduction Proof systems are built to prove statements. Proof systems are an inference machine with special statements, called provable state- ments being its final products. The starting points of the inference are called axioms of the system . We distinguish two kinds of axioms: logi- cal AL and specific SP . 1 Semantical link : we usually build a proof system for a given language and its seman- tics i.e. for a logic defined semantically. First step : we choose as a set of logical ax- ioms AL some subset of tautologies, i.e. statements always true. A proof system with only logical axioms AL is called a logic proof system . Building a proof system for which there is no known semantics we think about the logi- cal axioms as statements universally true. 2 We choose as axioms a finite set the state- ments we for sure want to be universally true, and whatever semantics follows they must be tautologies with respect to it. Logical axioms are hence not only tautolo- gies under an established semantics, but they also guide us how to establish a se- mantics, when it is yet unknown. The specific axioms SP are these formulas of the language that describe our knowl- edge of a universe we want to prove facts about. Specific axioms are not universally true, they are true only in the universe we are inter- ested to describe and investigate. 3 A proof system with logical axioms AL and specific axioms SP is called a formal the- ory . The inference machine is defined by a finite set of rules, called inference rules . The inference rules describe the way we are allowed to transform the information within the system with axioms as a staring point. The transformation process is called a for- mal proof and can be depicted as follows: AXIOMS RULES applied to AXIOMS Provable formulas RULES applied to any expressions above NEW Provable formulas . ..... . etc. ..... . The provable formulas are those for which we have a formal proof are called con- sequences of the axioms. 4 Semantical link : the rules have to preserve the truthfulness of what we are proving. Rules with this property are called sound rules and the system a sound proof system . Soundness Theorem : for any formula A of the language of the system S , If a formula A is provable in a logic proof system S , then A is a tautology. Formal theory with specific axioms SP , based on a logic defined by the axioms AL is a proof system S with logical axioms AL and specific axioms SP . Notation : TH S ( SP ). 5 Soundness Theorem for formal theory says: for any formula A of the language of the theory TH S ( SP ), If a formula A is provable in the theory TH S ( SP ) , then A is true in any model of the set of specific axioms SP ....
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## This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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7slides - Chapter 7: Proof Systems: Soundness and...

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